A gasp is the sound made by a sharp inward breath. After running up a steep hill, your breath will come in gasps.
When you make the sound of a gasp, you gasp: "It was so surprising to see my lost cat sitting calmly on the front porch that it made me gasp." People tend to gasp when they're shocked or hurt. A "last gasp" is the moment just before someone dies — it can also mean the point of completing or ending something: "It was the last gasp of video stores in my city when that one closed."